Dynamical systems and number theory

  • Date:
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
  • Lecturer: Matthew Palmer
  • Contact person: Reza Mohammadpour
  • Seminarium

Title: Vaaler's theorem in number fields

The seminar will take place at Ångström Laboratory, but it will also be possible to attend on Zoom. For more information, contact Reza Mohammadpour.

Abstract: 

Classical Diophantine approximation concerns the approximation of real numbers by rational numbers: in particular, with how well real numbers can be approximated by rational numbers. A long-standing conjecture in Diophantine approximation was the Duffin-Schaeffer conjecture, stated in 1941; this result was recently (2020) proven by Dimitris Koukoulopoulos and James Maynard. 

An area which is gaining more attention recently concerns generalisations of the notions of Diophantine approximation to other spaces in which we have a dense subfield with some sort of fractional structure. In this talk, I will discuss Diophantine approximation in number fields, where we use elements of an algebraic number field to simultaneously approximate in all of the number field's embeddings; I will state a version of a result known as Vaaler's theorem (a precursor to Koukoulopoulos and Maynard's theorem), give the rough structure of the proof in the classical case, and discuss how these elements generalise to the number fields case.